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Theorem suplocexprlemlub 7722
Description: Lemma for suplocexpr 7723. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (šœ‘ ā†’ āˆƒš‘„ š‘„ āˆˆ š“)
suplocexpr.ub (šœ‘ ā†’ āˆƒš‘„ āˆˆ P āˆ€š‘¦ āˆˆ š“ š‘¦<P š‘„)
suplocexpr.loc (šœ‘ ā†’ āˆ€š‘„ āˆˆ P āˆ€š‘¦ āˆˆ P (š‘„<P š‘¦ ā†’ (āˆƒš‘§ āˆˆ š“ š‘„<P š‘§ āˆØ āˆ€š‘§ āˆˆ š“ š‘§<P š‘¦)))
suplocexpr.b šµ = āŸØāˆŖ (1st ā€œ š“), {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢}āŸ©
Assertion
Ref Expression
suplocexprlemlub (šœ‘ ā†’ (š‘¦<P šµ ā†’ āˆƒš‘§ āˆˆ š“ š‘¦<P š‘§))
Distinct variable groups:   š‘¦,š“,š‘§   š‘„,š“,š‘¦   š‘§,šµ   šœ‘,š‘¦,š‘§   šœ‘,š‘„
Allowed substitution hints:   šœ‘(š‘¤,š‘¢)   š“(š‘¤,š‘¢)   šµ(š‘„,š‘¦,š‘¤,š‘¢)

Proof of Theorem suplocexprlemlub
Dummy variable š‘  is distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . 4 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ š‘¦<P šµ)
2 ltrelpr 7503 . . . . . . . 8 <P āŠ† (P Ɨ P)
32brel 4678 . . . . . . 7 (š‘¦<P šµ ā†’ (š‘¦ āˆˆ P āˆ§ šµ āˆˆ P))
43simpld 112 . . . . . 6 (š‘¦<P šµ ā†’ š‘¦ āˆˆ P)
54adantl 277 . . . . 5 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ š‘¦ āˆˆ P)
63simprd 114 . . . . . 6 (š‘¦<P šµ ā†’ šµ āˆˆ P)
76adantl 277 . . . . 5 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ šµ āˆˆ P)
8 ltdfpr 7504 . . . . 5 ((š‘¦ āˆˆ P āˆ§ šµ āˆˆ P) ā†’ (š‘¦<P šµ ā†” āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ))))
95, 7, 8syl2anc 411 . . . 4 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ (š‘¦<P šµ ā†” āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ))))
101, 9mpbid 147 . . 3 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))
11 simprrr 540 . . . . . 6 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ š‘  āˆˆ (1st ā€˜šµ))
12 suplocexpr.b . . . . . . . . . 10 šµ = āŸØāˆŖ (1st ā€œ š“), {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢}āŸ©
1312fveq2i 5518 . . . . . . . . 9 (1st ā€˜šµ) = (1st ā€˜āŸØāˆŖ (1st ā€œ š“), {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢}āŸ©)
14 npex 7471 . . . . . . . . . . . . 13 P āˆˆ V
1514a1i 9 . . . . . . . . . . . 12 (šœ‘ ā†’ P āˆˆ V)
16 suplocexpr.m . . . . . . . . . . . . 13 (šœ‘ ā†’ āˆƒš‘„ š‘„ āˆˆ š“)
17 suplocexpr.ub . . . . . . . . . . . . 13 (šœ‘ ā†’ āˆƒš‘„ āˆˆ P āˆ€š‘¦ āˆˆ š“ š‘¦<P š‘„)
18 suplocexpr.loc . . . . . . . . . . . . 13 (šœ‘ ā†’ āˆ€š‘„ āˆˆ P āˆ€š‘¦ āˆˆ P (š‘„<P š‘¦ ā†’ (āˆƒš‘§ āˆˆ š“ š‘„<P š‘§ āˆØ āˆ€š‘§ āˆˆ š“ š‘§<P š‘¦)))
1916, 17, 18suplocexprlemss 7713 . . . . . . . . . . . 12 (šœ‘ ā†’ š“ āŠ† P)
2015, 19ssexd 4143 . . . . . . . . . . 11 (šœ‘ ā†’ š“ āˆˆ V)
21 fo1st 6157 . . . . . . . . . . . . 13 1st :Vā€“ontoā†’V
22 fofun 5439 . . . . . . . . . . . . 13 (1st :Vā€“ontoā†’V ā†’ Fun 1st )
2321, 22ax-mp 5 . . . . . . . . . . . 12 Fun 1st
24 funimaexg 5300 . . . . . . . . . . . 12 ((Fun 1st āˆ§ š“ āˆˆ V) ā†’ (1st ā€œ š“) āˆˆ V)
2523, 24mpan 424 . . . . . . . . . . 11 (š“ āˆˆ V ā†’ (1st ā€œ š“) āˆˆ V)
26 uniexg 4439 . . . . . . . . . . 11 ((1st ā€œ š“) āˆˆ V ā†’ āˆŖ (1st ā€œ š“) āˆˆ V)
2720, 25, 263syl 17 . . . . . . . . . 10 (šœ‘ ā†’ āˆŖ (1st ā€œ š“) āˆˆ V)
28 nqex 7361 . . . . . . . . . . 11 Q āˆˆ V
2928rabex 4147 . . . . . . . . . 10 {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢} āˆˆ V
30 op1stg 6150 . . . . . . . . . 10 ((āˆŖ (1st ā€œ š“) āˆˆ V āˆ§ {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢} āˆˆ V) ā†’ (1st ā€˜āŸØāˆŖ (1st ā€œ š“), {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢}āŸ©) = āˆŖ (1st ā€œ š“))
3127, 29, 30sylancl 413 . . . . . . . . 9 (šœ‘ ā†’ (1st ā€˜āŸØāˆŖ (1st ā€œ š“), {š‘¢ āˆˆ Q āˆ£ āˆƒš‘¤ āˆˆ āˆ© (2nd ā€œ š“)š‘¤ <Q š‘¢}āŸ©) = āˆŖ (1st ā€œ š“))
3213, 31eqtrid 2222 . . . . . . . 8 (šœ‘ ā†’ (1st ā€˜šµ) = āˆŖ (1st ā€œ š“))
3332eleq2d 2247 . . . . . . 7 (šœ‘ ā†’ (š‘  āˆˆ (1st ā€˜šµ) ā†” š‘  āˆˆ āˆŖ (1st ā€œ š“)))
3433ad2antrr 488 . . . . . 6 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ (š‘  āˆˆ (1st ā€˜šµ) ā†” š‘  āˆˆ āˆŖ (1st ā€œ š“)))
3511, 34mpbid 147 . . . . 5 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ š‘  āˆˆ āˆŖ (1st ā€œ š“))
36 suplocexprlemell 7711 . . . . 5 (š‘  āˆˆ āˆŖ (1st ā€œ š“) ā†” āˆƒš‘§ āˆˆ š“ š‘  āˆˆ (1st ā€˜š‘§))
3735, 36sylib 122 . . . 4 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ āˆƒš‘§ āˆˆ š“ š‘  āˆˆ (1st ā€˜š‘§))
38 simprl 529 . . . . . . . . 9 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ š‘  āˆˆ Q)
3938ad2antrr 488 . . . . . . . 8 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘  āˆˆ Q)
40 simprrl 539 . . . . . . . . 9 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ š‘  āˆˆ (2nd ā€˜š‘¦))
4140ad2antrr 488 . . . . . . . 8 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘  āˆˆ (2nd ā€˜š‘¦))
42 simpr 110 . . . . . . . 8 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘  āˆˆ (1st ā€˜š‘§))
43 rspe 2526 . . . . . . . 8 ((š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜š‘§))) ā†’ āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)))
4439, 41, 42, 43syl12anc 1236 . . . . . . 7 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)))
454ad4antlr 495 . . . . . . . 8 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘¦ āˆˆ P)
4619ad4antr 494 . . . . . . . . 9 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š“ āŠ† P)
47 simplr 528 . . . . . . . . 9 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘§ āˆˆ š“)
4846, 47sseldd 3156 . . . . . . . 8 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘§ āˆˆ P)
49 ltdfpr 7504 . . . . . . . 8 ((š‘¦ āˆˆ P āˆ§ š‘§ āˆˆ P) ā†’ (š‘¦<P š‘§ ā†” āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜š‘§))))
5045, 48, 49syl2anc 411 . . . . . . 7 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ (š‘¦<P š‘§ ā†” āˆƒš‘  āˆˆ Q (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜š‘§))))
5144, 50mpbird 167 . . . . . 6 (((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) āˆ§ š‘  āˆˆ (1st ā€˜š‘§)) ā†’ š‘¦<P š‘§)
5251ex 115 . . . . 5 ((((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) āˆ§ š‘§ āˆˆ š“) ā†’ (š‘  āˆˆ (1st ā€˜š‘§) ā†’ š‘¦<P š‘§))
5352reximdva 2579 . . . 4 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ (āˆƒš‘§ āˆˆ š“ š‘  āˆˆ (1st ā€˜š‘§) ā†’ āˆƒš‘§ āˆˆ š“ š‘¦<P š‘§))
5437, 53mpd 13 . . 3 (((šœ‘ āˆ§ š‘¦<P šµ) āˆ§ (š‘  āˆˆ Q āˆ§ (š‘  āˆˆ (2nd ā€˜š‘¦) āˆ§ š‘  āˆˆ (1st ā€˜šµ)))) ā†’ āˆƒš‘§ āˆˆ š“ š‘¦<P š‘§)
5510, 54rexlimddv 2599 . 2 ((šœ‘ āˆ§ š‘¦<P šµ) ā†’ āˆƒš‘§ āˆˆ š“ š‘¦<P š‘§)
5655ex 115 1 (šœ‘ ā†’ (š‘¦<P šµ ā†’ āˆƒš‘§ āˆˆ š“ š‘¦<P š‘§))
Colors of variables: wff set class
Syntax hints:   ā†’ wi 4   āˆ§ wa 104   ā†” wb 105   āˆØ wo 708   = wceq 1353  āˆƒwex 1492   āˆˆ wcel 2148  āˆ€wral 2455  āˆƒwrex 2456  {crab 2459  Vcvv 2737   āŠ† wss 3129  āŸØcop 3595  āˆŖ cuni 3809  āˆ© cint 3844   class class class wbr 4003   ā€œ cima 4629  Fun wfun 5210  ā€“ontoā†’wfo 5214  ā€˜cfv 5216  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   <Q cltq 7283  Pcnp 7289  <P cltp 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-qs 6540  df-ni 7302  df-nqqs 7346  df-inp 7464  df-iltp 7468
This theorem is referenced by:  suplocexpr  7723
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